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When can we say we fully understand QCD?

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What constitutes a sufficient "understanding" of QCD? Take an analogy, if an undergrad understands Newton's three laws well, the formula for Newtonian gravity and know how to solve for conic sectional orbits, then I won't cringe if this undergrad claims to have understood Newtonian gravity.

Now, I'd like to ask what makes people say we haven't understood low energy QCD yet: if it means we must have a systematic nonperturbative computational scheme to calculate everything of interest, isn't lattice QCD (in principle) enough? If it means we must have a rigorous mathematical foundation for it, then in the Newtonian analogy, wouldn't that imply no physicist understood Newtonian gravity before mathematical analysis was discovered, which is an absurd statement?

Still, my impression is that people talk about QCD as if some fundamentally new enlightenment is needed, what's the reason for that? What do we really hope to accomplish so that a complete "understanding" can be claimed? How do we know we are not just pushing the technical boundary further and further?

Like we know the formular for newton's law of mechanics, But does that mean you can compute the precise orbits of 9 planets accurate to 1000 years? You would simply put it in a computer, if you had to do that kind of a calculation.

I don't know of anyone who seriously thinks he can solve a 9 body problem analytically. There is probably very little hope, unless ofcourse someone comes along and actually does solve it.

It seems to me that there is some hope with QCD to say atleast something more analytically. Yang mills mass gap conjucture for instance is one way to motivate research in that direction. I think the problem is strictly of a mathematical nature.

However there are other related reseach ideas, especially in understanding the supersymmetric versions of QCD to provide deeper insight into the structure of physics itself.

Like we know the formular for newton's law of mechanics, But does that mean you can compute the precise orbits of 9 plants accurate to 1000 years? You would simply put it in a computer, if you had to do that kind of a calculation.

What do you want to convey by this paragraph? Not knowing how to do a calculation that precise doesn't mean we don't understand Newtonian gravity.

It seems to me that there is some hope with QCD to say atleast something more analytically. Yang mills mass gap conjucture for instance is one way to motivate research in that direction. I think the problem is strictly of a mathematical nature.

But again in the Newtonian analogy, could mass gap problem be like the problem of stability of the solar system? It's also strictly of mathematical nature, and heavy machinery of analytic nature is probably crucial, but not understanding it seems to pose no big threat to the claim that we understand Newtonian gravity. I admit I'm partly struggling with the philosophical underpinning of "understanding". I need to go to bed after writing this comment, don't hold your breath for my next reply:-)

The mass gap is a statement about the smallest mass bound state of the theory, corresponding in the solar analogy to the Kepler problem, which can be solved exactly and hence is easy to understand. Whereas the stability of the solar system is the analogue of the question of whether a particular set of particles with given properties (sun and planets) form a bound state. Clearly understanding the latter is far more demanding than understanding the former.

"as if some fundamentally new enlightenment is needed" : yes, always. Nothing is definitive. The chain to the ultimate questions never ends.

For QCD, at least, there is the question of an astronomically interesting 4th neutrino and since 2012, 3 very different experiments report weird behaviors of muons compared to electrons, like if they were differering not only by the mass. The latter is not yet completely confirmed but it shows that the physicists rely only on their measures devices :) From a GUT point of view, the standard model is not what symmetries friends might dream.

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The understanding of an ordinary differential equation has nothing to do with being able to successfully execute a Runge-Kutta method. The latter only gives numerical values for an individual trajectory (or if called multiple times, for many).

But understanding means to know where its fixed points are, how it behaves for large times, how sensitive the solutions are to a change of initial conditions, etc.. Not single numbers or curves but the general pattern of arbitrary solutions.

We are far for such an understanding of QCD except in the very high energy region. Thus people say that QCD is not understood because in the infrared domain (confinement, mass gap, bound states) we have very little grasp on how to obtain properties of arbitrary solutions at a level that conveys more than individual numerical numbers. Lattice QCD is just a black box that spits out a (fairly inaccurate) number for every numerical question we ask, it give not understanding in any sense.

One can say we understand QCD if we can derive from its action the low energy Hamiltonian and the bound state content (mesons, baryons, and perhaps glueballs) to an extent that we can match the meson and baryon data from the Particle Data Group to its particle spectrum.

We wouldn't understand Newtonian gravity if we coudn't do a qualitative analysis of the 3-body problem. There is no analytic solution but still we understand (and can approximate) essentially everything about its behavior, independent of the detailed parameters of the problem.

Lattice calculations in the 3-body problem would correspond to discretizing the dynamics using second-order divided differences, which very poorly resolves the dynamics of a 3-body problem, so a very fine lattice would be required to give good results over a significant time span, and then it would just be for a single system - nothing general.

Perturbation theory around a 2-body problem gives very useful analytic approximations not just for a single system but for all problems in the class, and one can deduce a lot fronm its sutdy, whereas even a better discretization method (like modern symplectic integrators) give just a single trajectory, or if repeated a bunch of trajectories, from which one cannot deduce much about the qualitative behavior.

Understanding always means to be able to derive qualitative understanding, not just numbers. (And even the numbers obtained from lattice QCD are not impressive. I haven't seen even a single attempt to compute the full baryon and meson spectrum from QCD. Given that QCD needs no numerical input to define it, the accuracy for basic predictions such as the proton mass (by lattice QCD or by Schwinger-Dyson equations) is perhaps 5 percent, and it will not grow much even if the speed of computers and algorithms increase by a factor of $10^6$ (which is not realistic). No, we need a much better understanding!

We understand QED much better than QCD, because there are many good approximation schemes which give qualitative information about almost everything of interest. But even QED is not completely understood as we don't have a logically satisfying setting for the theory, and things like the nonperturbative existence of the Landau pole are unsettled.

Hi Arnold, thanks for all the valuable input. I'm aware it's usually too exacting to envision what's the next big thing(or if there is any) before a big thing actually emerges, but I'm currently facing a choice problem between going into more formal QFT/string or going into more down-to-earth QCD study for a PhD, so I can't help thinking about these "big-picture" issues, which sometimes can simply be ill-posed(but I certainly hope it hasn't been the case for this question). Let me chew on the issue for some more time.

@JiaYiyang: I think there are many unsolved issues in QCD (and some even in QED). It is a pity that many of the best minds go to a more speculative side of theoretical physics such as string theory rather than work on conventional QFT. I believe that standard QFT remains valid even at the Planck scale and below, and that the real progress in fundamental physics will come from getting a stronger nonperturbative analytic grasp on QFT - e.g., through finding a valid Hilbert space setting - rather than from changing the foundations. (Of course, this doesn't necessarily affect the choice of a Ph.D. topic, as this must more be something tractable rather than something aimed at the physics of the future.)

A lot of intuition to understanding standard non perturbative QFT arises from string theory. Most of the string theorist are primarily experts in QFT. Although a lot of work is being done in understanding string theory on its own merit a lot of work is being done in understanding QFT in general using tools from string theory. I think that at this time only using string tools we can get a better grasp into QFT. Tools like non-perturbative dualities, AdS/CFT and so on.

@ArnoldNeumaier: D-branes are one of the most important objects in string theory. The whole AdS/CFT and its extensions are based in completely geometrical constructions using D-branes. Indeed, QCD is still far from being understood via a holographic dual description, but this, nonetheless, is a very promising area of research and one needs to understand to some extent string theory in order to understand the various constructions. In any case most of the current string theory research that is not focused on phenomenology it is focused on understanding some sort of dual field theories.

@ArnoldNeumaier I will not pretend I know a lot about the models of holographic QCD but in principle you do calculations on the bulk where SUGRA is present. In that sense was my comment above. I agree, you do not need to understand a whole big deal of string theory to apply holography in some models and this is why a lot of CMT people have shifted in holography. To create new ones that would be better and better approximations to various kinds of dual field theories though, you need to understand very well string theory and the various geometric constructions.

@conformal_gk: Nothing in AdS/CFT as applied to QCD uses the notion of a string. Of course string theorists develop a lot of QFT along the side, but the tools they develop for the latter are QFT tools, not string theory tools. Only the motivation for having developed them comes from string theory.

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We will understand completely QCD when we are able to compute any quantity we want at any energy scale we want. The main object of interest are of course $n$-point correlation functions in the IR below $\Lambda_{\text{QCD}}$. We also need to understand QCD instantons, QCD phase transitions like the transition to the quark-gluon plasma. Also, an open problem is the mathematical proof of the existence of the Yang-Mills (e.g. QCD) mass gap. The problem with the lattice is that it is Euclidean and that you solve numerically. Understanding the theory and being able to predict requires in some sense analytical and exact results. Lattice as well as other approaches give you intuition but this does not mean you have solved the theory.

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